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Dynamical low-rank approximation strategies for nonlinear feedback control problems

  • Luca Saluzzi ORCID logo EMAIL logo und Maria Strazzullo
Veröffentlicht/Copyright: 7. November 2025
Journal of Numerical Mathematics
Aus der Zeitschrift Journal of Numerical Mathematics

Abstract

This paper addresses the stabilization of dynamical systems in the infinite horizon optimal control setting using nonlinear feedback control based on State-Dependent Riccati Equations (SDREs). While effective, the practical implementation of such feedback strategies is often constrained by the high dimensionality of state spaces and the computational challenges associated with solving SDREs, particularly in parametric scenarios. To mitigate these limitations, we introduce the Dynamical Low-Rank Approximation (DLRA) methodology, which provides an efficient and accurate framework for addressing high-dimensional feedback control problems. DLRA dynamically constructs a low-dimensional representation that evolves with the problem, enabling the simultaneous resolution of multiple parametric instances in real-time. We propose two novel algorithms to enhance numerical performances: the cascade-Newton–Kleinman method and Riccati-based DLRA (R-DLRA). The cascade-Newton–Kleinman method accelerates convergence by leveraging Riccati solutions from the nearby parameter or time instance, supported by a theoretical sensitivity analysis. R-DLRA integrates Riccati information into the DLRA basis construction to improve the quality of the solution. These approaches are validated through nonlinear one-dimensional and two-dimensional test cases showing transport-like behavior, demonstrating that R-DLRA outperforms standard DLRA and Proper Orthogonal Decomposition-based model order reduction in both speed and accuracy, offering a superior alternative to Full Order Model solutions.

Keywords: model order reduction; dynamical low-rank approximation; parametric nonlinear feedback control; state-dependent Riccati equation; Newton–Kleinman method
MSC 2010 Classification: 49M41; 76D55; 93C20
Corresponding author: Luca Saluzzi, Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro, 5, Roma, 00185, Italy; and INDAM, GNCS Member, Roma , Italy, E-mail: 

Funding source: Metodi di riduzione di modello ed approssimazioni di rango basso per problemi alto-dimensionali

Award Identifier / Grant number: CUP E53C23001670001 (Metodi di riduzione di modello ed approssimazioni di rango basso per problemi alto-dimensionali)

Funding source: European Union – Next Generation EU (PRIN 2022 program (D.D. 104 - 02/02/2022 Ministero dell’Università e della Ricerca))

Award Identifier / Grant number: 20227K44ME (Full and Reduced order modelling of coupled systems: focus on nonmatching methods and automatic learning (FaReX))

Funding source: European Union’s Horizon 2020 (Marie Sk lodowska-Curie Actions Grant)

Award Identifier / Grant number: 872442 (ARIA)

Funding source: ECCOMAS EYIC Grant

Acknowledgement

The authors express their sincere gratitude for the many insightful discussions with Stefano Massei and Cecilia Pagliantini.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission. All authors contributed to the design and implementation of the research, to the analysis of the results and to the writing of the manuscript.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: We acknowledge the INdAM - GNCS Project “Metodi di riduzione di modello ed approssimazioni di rango basso per problemi alto-dimensionali” (CUP E53C23001670001). MS thanks the “20227K44ME - Full and Reduced order modelling of coupled systems: focus on nonmatching methods and automatic learning (FaReX)” project – funded by European Union – Next Generation EU within the PRIN 2022 program (D.D. 104 - 02/02/2022 Ministero dell’Università e della Ricerca). This manuscript reflects only the authors’ views and opinions and the Ministry cannot be considered responsible for them. MS acknowledges the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Actions, grant agreement 872442 (ARIA). MS thanks the ECCOMAS EYIC Grant “CRAFT: Control and Reg-reduction in Applications for Flow Turbulence”.

  7. Data availability: The data that support the findings of this study are openly available at https://github.com/saluzzi/DLRA-Feedback-Control.

Appendix A: Additional results on the sensitivity estimates

In this appendix, we collect all the results needed to prove and validate the estimate of Section 3. Section A.1 details the propositions necessary to the proofs of Theorems 2 and 3. In Section A.2, we propose a numerical example validating the theoretical sensitivity analysis.

A.1 Auxiliary results

Proposition 3.

Given a vector field f(t, y) satisfying the one-sided Lipschitz condition (10), then the following estimate holds

(44) ‖ y ( t ; y 0 ) − y ( t ; y ̃ 0 ) ‖ ⩽ ‖ y 0 − y ̃ 0 ‖ e L t .

Proof.

Let us denote by y 1(t) and y 2(t) the solutions of the ODE with corresponding initial condition y 0 and y ̃ 0 and define Δy(t) = y 1(t) − y 2(t). Then we have

Δ y ( t ) ⋅ d d t Δ y ( t ) = Δ y ( t ) ⋅ ( f ( t , y 1 ( t ) ) − f ( t , y 2 ( t ) ) ) .

Using the one-sided Lipschitz condition:

Δ y ( t ) ⋅ ( f ( t , y 1 ( t ) ) − f ( t , y 2 ( t ) ) ) ⩽ L ‖ Δ y ( t ) ‖ 2 .

Since d d t ‖ Δ y ( t ) ‖ 2 = 2 Δ y ( t ) ⋅ d d t Δ y ( t ) , we have

d d t ‖ Δ y ( t ) ‖ 2 ⩽ 2 L ‖ Δ y ( t ) ‖ 2 .

By Gronwall’s inequality, we obtain the result. □

Proposition 4.

Given Assumptions 2 (a)–(c), the vector field f(t, y) = A cl(y)y satisfies the one-sided Lipschitz condition (10) in a domain Ω with constant

L = max x ∈ Ω logm ( A cl ( x ) ) + L cl max y ∈ Ω ‖ y ‖ .

Proof.

We aim to verify if the function A cl(y)y satisfies condition (10). Let us manipulate the left-hand side of condition (10) to obtain

( x − y ) ⊤ ( A cl ( x ) x − A cl ( y ) y ) = ( x − y ) ⊤ A cl ( x ) ( x − y ) + ( x − y ) ⊤ ( A cl ( x ) − A cl ( y ) ) y .

By the definition of logarithmic norm we obtain

( x − y ) ⊤ A cl ( x ) ( x − y ) ⩽ logm ( A cl ( x ) ) ‖ x − y ‖ 2 ,

while by Assumption 2 (a)

| ( x − y ) ⊤ ( A cl ( x ) − A cl ( y ) ) y | ⩽ L cl ‖ y ‖ ‖ x − y ‖ 2 .

Considering the maximum in x and y over the domain Ω gives the result. □

Corollary 1.

Given Assumptions 2 (a)–(d), the following estimate holds

‖ y ( t ; y 0 ( μ ) ) − y ( t ; y 0 ( μ ̃ ) ) ‖ ⩽ L y 0 ‖ μ − μ ̃ ‖ e L t

with L = max x∈Ωlogm(A cl(x)) + L cl max y∈Ω‖y‖.

Proof.

By Proposition 4, we know that the function f(t, y) = A cl(y)y satisfies the one-sided Lipschitz condition (10). Applying Proposition 3 and Assumption 2 (d) we obtain the result. □

Proposition 5.

Given Assumption 2 (f), then ‖A cl(y(t))y(t)‖ is decreasing in time.

Proof.

Denoting z(t) = A cl(y(t))y(t), we have

d d t ‖ z ( t ) ‖ 2 = 2 z ( t ) ⊤ ( ∇ A cl ( y ( t ) ) ) y ( t ) + A ( y ( t ) ) z ( t ) ⩽ 2 ( logm ( A ( y ) ) + logm ( ( ∇ A cl ( y ) ) y ) ) ‖ z ( t ) ‖ 2 ,

and the result comes from Assumption 2 (f). □

A.2 A numerical validation

In this section, we want to show how the theoretical findings derived in Section 3 can be applied in practice. Let us consider the optimal control problem driven by the following nonlinear reaction–diffusion equation:

(45) d d t y ( t , x ) = σ ∂ x x y ( t , x ) + y ( t , x ) 2 + u ( t , x ) , ( t , x ) ∈ R + × [ 0,2 ]

with associated cost functional

J ( u , y 0 ) = ∫ 0 + ∞ ∫ Ω | y ( t , x ) | 2 d x + ∫ Ω | u ( t , x ) | 2 d x d t .

Fixing σ = 10−3 and performing a semidiscretization via finite difference with N h = 50 grid points, we obtain

y ̇ ( t ) = A ( y ) y ( t ) + u ( t ) ,

where

A ( y ) = σ A 0 + diag ( y ◦ y )

with A 0 arising from the discretization of the Laplacian with Neumann boundary conditions, ◦ stands for the Hadamard product and diag(v) indicates a diagonal matrix with the components of the vector v on the main diagonal.

Let us consider a set of parameterized initial conditions

y 0 ( x ; μ ) = μ ⁡ sin ( π x ) , μ ∈ [ 1,10 ] .

We note that in this case A(y) is symmetric for all y ∈ R d and the term F is an identity matrix multiplied by a scalar depending on the space discretization. As a result, the closed-loop matrix is symmetric, and all of its eigenvalues are real. Let us denote by λ μ 1 , μ 2 , t 1 , t 2 the maximum eigenvalue of the matrix A μ 2 , t 2 − F P μ 1 , t 1 , i.e., the starting closed-loop matrix for the parameter μ 2 at time t 2 using as initial guess P μ 1 , t 1 . Our aim is to verify whenever this value is negative and, hence, P μ 1 , t 1 represents a stabilizing initial guess. We proceed applying the following steps:

  1. we start at initial time from μ 1 = 1 and we solve the following SDRE

    A μ 1 , 0 ⊤ P μ 1 , 0 + P μ 1 , 0 A μ 1 , 0 − P μ 1 , 0 F P μ 1 , 0 = − Q

    using the Matlab function icare;

  2. we compute the approximate solution H μ 1 , 0 of the Lyapunov equation (16) by the Matlab function lyap;

  3. we note that

    ‖ Δ A μ 1 , μ 2 , 0,0 ‖ 2 = | μ 1 2 − μ 2 2 | ‖ diag ( sin ( π x ) ) ‖ 2 ,

    where sin ( π x ) = [ sin ( π x 1 ) , … , sin ( π x d ) ] ⊤ ;

  4. condition (15) now reads

    | μ 1 2 − μ 2 2 | < c = 1 4 ‖ H μ 1 , 0 ‖ 2 ‖ diag ( sin ( π x ) ) ‖ 2 ,

    where the right-hand side is equal to 1.0011 in this case. Hence, condition (15) is satisfied if

    Δ μ < − 1 + 1 + c ≈ 0.4146 .

If we consider Δμ = 0.45, the condition (15) is not satisfied and we note that the maximum eigenvalue of the closed loop matrix A μ 2 , 0 − F P μ 1 , 0 is positive ( λ μ 1 , μ 2 , 0,0 = 0.0019 ) , hence P μ 1 , 0 is not a stabilizing initial guess. On the other hand, considering Δμ = 0.4, as expected, P μ 1 , 0 is a stabilizing initial guess ( λ μ 1 , μ 2 , 0,0 = − 0.1342 ). This provides a strategy to compute the parameter stepsize to ensure the cascade approach at the initial time.

Now let us consider the time integration with final time T = 0.1 with Δμ = 0.1. In the top-left panel of Figure 10, we compare the behavior in time of the terms g 1 ( t ) = ‖ Δ A μ , μ ̃ , t , t ‖ (denoted in the plot as Lower Estimate), g 2 ( t ) = L A L y 0 ‖ μ − μ ̃ ‖ e L t (One-sided Upper Estimate) and g 3 ( t ) = 1 2 ‖ H μ , t ‖ (Upper Estimate). The constants L and L A can be estimated along the trajectories in the following way

L A = max i ‖ A ( y μ 1 , t i ) − A ( y μ 2 , t i ) ‖ ‖ y μ 1 , t i − y μ 2 , t i ‖ ,

L = max i ( A cl ( y μ 1 , t i ) − A cl ( y μ 2 , t i ) ) ⊤ ( y μ 1 , t i − y μ 2 , t i ) ‖ y μ 1 , t i − y μ 2 , t i ‖ .

We note that the upper estimate exhibits an almost constant behavior, remaining close to values around 1. On the other hand, the Lower Estimate starts below as expected, since the initial choice for Δμ satisfies the condition (15). The estimated one-sided Lipschitz constant L is negative and equal to −1.9897, reflecting the decreasing behavior of the One-sided Upper Estimate and of the Lower Estimate. Condition (19) is depicted in the top-right panel of Figure 10. The right-hand side of the estimate (19) is decreasing, but always positive, while the one-sided Lipschitz constant L is negative. Hence, condition (19) is satisfied for the entire time interval [0, 0.1]. From this, we can infer that P μ 1 , t consistently serves as a stabilizing initial guess. This is demonstrated in the bottom panel of Figure 10, where the maximum eigenvalue remains negative throughout the time interval. Let us now perform a similar analysis for the time step Δt. For simplicity, we will use the explicit Euler method for integration. Given the initial condition y μ 1 , 0 = sin ( π x ) , consider the update y μ 1 , Δ t = y μ 1 , 0 + Δ t ( A ( y μ 1 , 0 ) − F P μ 1 , 0 ) y μ 1 , 0 . We define f cl,0 as f c l , 0 = ( A ( y μ 1 , 0 ) − F P μ 1 , 0 ) y μ 1 , 0 . Now, we compute

‖ Δ A μ 1 , μ 1 , 0 , Δ t ‖ 2 = ‖ diag ( y μ 1 , Δ t ◦ y μ 1 , Δ t ) − diag ( y μ 1 , 0 ◦ y μ 1 , 0 ) ‖ = ( Δ t ) 2 f c l , 0 2 + 2 Δ t f c l , 0 ⋅ y μ 1 , 0 ,

then condition (15) becomes

( Δ t ) 2 ‖ f c l , 0 ‖ 2 + 2 Δ t f c l , 0 ⋅ y μ 1 , 0 − 1 2 ‖ H μ 1 , 0 ‖ 2 < 0 ,

which leads to the solution

Δ t < − f c l , 0 ⋅ y μ 1 , 0 + ( f c l , 0 ⋅ y μ 1 , 0 ) 2 + 1 2 ‖ H μ 1 , 0 ‖ 2 ‖ f c l , 0 ‖ 2 ≈ 1.5694 .

In fact, for Δt = 1.6, the maximum eigenvalue of the closed loop matrix A ( y μ 1 , Δ t ) − F P μ 1 , 0 is positive ( λ μ 1 , μ 1 , 0 , Δ t = 0.8752 ) , whereas for Δt = 1.5, it is negative ( λ μ 1 , μ 1 , 0 , Δ t = − 0.7858 ) . This section justifies the use of the cascade information in solving the SDRE by means of NK.

Figure 10: Top: Verification of the estimate (15) (left) and the estimate (19) (right). Bottom: Maximum eigenvalue λ μ 1 , μ 2 , t , t in the time interval [0, 0.1].
Figure 10:

Top: Verification of the estimate (15) (left) and the estimate (19) (right). Bottom: Maximum eigenvalue λ μ 1 , μ 2 , t , t in the time interval [0, 0.1].

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Received: 2025-01-13
Accepted: 2025-09-08
Published Online: 2025-11-07

© 2025 Walter de Gruyter GmbH, Berlin/Boston

https://doi.org/10.1515/jnma-2025-0005
Schlagwörter für diesen Artikel
model order reduction; dynamical low-rank approximation; parametric nonlinear feedback control; state-dependent Riccati equation; Newton–Kleinman method
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Heruntergeladen am 26.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jnma-2025-0005/pdf?lang=de
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